Compactness in Ginzburg-Landau Energy by Kinetic Averaging

We consider a Ginzburg-Landau energy for two dimensional divergence free fields appearing 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(). Our proof is based on a kinetic interpretation of the entropies which were introduced by Desimone, Kohn, Müller and Otto. The so-called kinetic averaging lemmas allow to generalize their compactness results. Also the method yields a kinetic equation for the limit where the righthand 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 level of singularities in the limit. c © 2001 John Wiley & Sons, Inc.