# Convergence of the parabolic Ginzburg–Landau equation to motion by mean curvature

### Abstract

For the complex parabolic Ginzburg-Landau equation, we prove that, asymptotically, vorticity evolves according to motion by mean curvature in Brakke’s weak formulation. The only assumption is a natural energy bound on the initial data. In some cases, we also prove convergence to enhanced motion in the sense of Ilmanen.

## Authors

Fabrice Bethuel

Laboratoire J.-L. Lions
Université Pierre et Marie Curie
75013 Paris
France
and
Institut Universitaire de France
75005 Paris
France

Giandomenico Orlandi

Dipartimento di Informatica
Università di Verona
37129 Verona
Italy

Didier Smets

Laboratoire J.-L. Lions
Université Pierre et Marie Curie
75013 Paris
France