Vortex liquids and the Ginzburg-Landau equation

TL;DR

This paper analyzes vortex dynamics in the time-dependent Ginzburg-Landau equation, providing quantitative bounds and demonstrating convergence to hydrodynamic limits for large vortex systems without gauge fields.

Contribution

It establishes explicit convergence rates and bounds for vortex dynamics in the Ginzburg-Landau model with large vortex numbers and specific boundary conditions.

Findings

Proved convergence of vortex solutions to hydrodynamic limits.

Established quantitative bounds on kinetic energy and other quantities.

Demonstrated explicit convergence rates for dilute vortex liquids.

Abstract

We establish vortex dynamics for the time-dependent Ginzburg-Landau equation for asymptotically large numbers of vortices for the problem without a gauge field and either Dirichlet or Neumann boundary conditions. As our main tool, we establish quantitative bounds on several fundamental quantities, including the kinetic energy, that lead to explicit convergence rates. For dilute vortex liquids we prove that sequences of solutions converge to the hydrodynamic limit.