Limiting motion for the parabolic Ginzburg-Landau equation with infinite energy data

TL;DR

This paper investigates the evolution of vorticity in the parabolic Ginzburg-Landau equation with infinite energy initial data, showing convergence to mean curvature motion and analyzing vortex dynamics in various dimensions.

Contribution

It extends previous work to infinite energy data, demonstrating asymptotic vorticity behavior and vortex motion properties in higher dimensions and complex initial conditions.

Findings

Vorticity evolves according to mean curvature in weak formulation.

In the plane, point vortices remain stationary over the original time scale.

Results apply to infinite lattice and filament vortices.

Abstract

We study a class of solutions to the parabolic Ginzburg-Landau equation in dimension 2 or higher, with ill-prepared infinite energy initial data. We show that, asymptotically, vorticity evolves according to motion by mean curvature in Brakke's weak formulation. Then, we prove that in the plane, point vortices do not move in the original time scale. These results extend the work of Bethuel, Orlandi and Smets [8, 9] for infinite energy data, they allow to consider the point vortices on a lattice (in dimension 2), or filament vortices of infinite length (in dimension 3).