Structural descriptions of limits of the parabolic Ginzburg-Landau equation on closed manifolds

TL;DR

This paper characterizes the limiting behavior of solutions to the parabolic Ginzburg-Landau equation on closed manifolds, revealing a decomposition of energy measures and their evolution over time.

Contribution

It provides a detailed structural description of energy measure limits, including their decomposition and evolution, extending previous work to higher-dimensional closed manifolds.

Findings

Energy measures decompose into diffuse and concentrated parts

Diffuse part evolves via heat equation

Concentrated part follows Brakke flow

Abstract

In the setting of a compact Riemannian manifold of dimension $N\ge3$ we provide a structural description of the limiting behaviour of the energy measures of solutions to the parabolic Ginzburg-Landau equation. In particular, we provide a decomposition of the limiting energy measure into a diffuse part, which is absolutely continuous with respect to the volume measure, and a concentrated part supported on a codimension $2$ rectifiable subset. We also demonstrate that the time evolution of the diffuse part is determined by the heat equation while the concentrated part evolves according to a Brakke flow. This paper extends the work of Bethuel, Orlandi, and Smets.