Nearly Parallel Vortex Filaments in the 3D Ginzburg-Landau Equations

TL;DR

This paper develops a rigorous framework linking the Ginzburg-Landau energy to nearly parallel vortex filaments in 3D, providing new insights into filament clustering and higher-order asymptotics.

Contribution

It introduces a new functional describing vortex filament energy as a Gamma-limit, connecting Ginzburg-Landau theory with filament configurations in 3D.

Findings

Established the Gamma-limit functional for vortex filaments

Distinguished multiplicity one filaments from higher multiplicity clusters

Proved existence of solutions with filament clusters governed by the new functional

Abstract

We introduce a framework to study the occurrence of vortex filament concentration in $3D$ Ginzburg-Landau theory. We derive a functional that describes the free-energy of a collection of nearly-parallel quantized vortex filaments in a cylindrical $3$-dimensional domain, in certain scaling limits; it is shown to arise as the $\Gamma$-limit of a sequence of scaled Ginzburg-Landau functionals. Our main result establishes for the first time a long believed connection between the Ginzburg-Landau functional and the energy of nearly parallel filaments that applies to many mathematically and physically relevant situations where clustering of filaments is expected. In this setting it also constitutes a higher-order asymptotic expansion of the Ginzburg-Landau energy, a refinement over the arclength functional approximation. Our description of the vorticity region significantly improves on previous studies and enables us to rigorously distinguish a collection of multiplicity one vortex filaments from an ensemble of fewer higher multiplicity ones. As an application, we prove the existence of solutions of the Ginzburg-Landau equation that exhibit clusters of vortex filaments whose small-scale structure is governed by the limiting free-energy functional.