Compactness in Ginzburg-Landau energy by kinetic averaging

We consider a Ginzburg-Landau energy for two-dimensional, divergence-free fields, which appear in the gradient theory of phase transition, for instance. We prove that as the relaxation parameter vanishes, families of such fields with finite energy are compact in L^p(Ω). Our proof is based on a kinetic interpretation of the entropies that were introduced by DeSimone, Kohn, Müller, and Otto. The so-called kinetic averaging lemmas allow us to generalize their compactness results. Also, the method yields a kinetic equation for the limit where the right-hand side is an unknown kinetic defect bounded measure from which we deduce some Sobolev regularity. This measure also satisfies some cancellation properties depending on its local regularity, which seem to indicate several levels of singularities in the limit.