Generation of vortices in the Ginzburg-Landau heat flow

TL;DR

This paper studies the evolution of vortices in the Ginzburg-Landau heat flow on a 2D torus, demonstrating conservation of initial zeros and describing vortex dynamics in a logarithmic energy regime.

Contribution

It establishes the conservation of initial zeros and characterizes vortex evolution in the Ginzburg-Landau heat flow on a torus, connecting to existing vortex dynamics theories.

Findings

Zeros are conserved during the flow.

Flow enters a logarithmic energy regime rapidly.

Vortex evolution aligns with prior theoretical frameworks.

Abstract

We consider the Ginzburg-Landau heat flow on the two-dimensional flat torus, starting from an initial data with a finite number of nondegenerate zeros -- but possibly very high initial energy. We show that the initial zeros are conserved and the flow rapidly enters a logarithmic energy regime, from which the evolution of vortices can be described by the works of Bethuel, Orlandi and Smets.