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

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).