Compactness in Ginzburg-Landau energy by kinetic averaging

We consider a Ginzburg-Landau energy for two-dimensional, divergence-free f ields, which appear in the gradient theory of phase transition. for instanc e. Wt: prove that as the relaxation parameter vanishes, families of such fi elds with finite energy are compact in LP(Omega). Our proof is based on a k inetic interpretation of the entropies that were introduced by DeSimone, Ko hn, Muller, and Otto. The so-called kinetic averaging lemmas allow us to ge neralize their compactness results. Also, the method yields a kinetic equat ion for the limit where the right-hand side is an unknown kinetic defect bo unded measure from which we deduce some Sobolev regularity. This measure al so satisfies some cancellation properties depending on its local regularity , which seem to indicate several levels of singularities in the limit. (C) 2001 John Wiley & Sons, Inc.