Scaling limits and regularity results for a class of Ginzburg-Landau systems

We study a class of parabolic systems which includes the Ginzburg-Landau heat flow equation, u^∈_t - Δu^∈ + 1/∈²(|u^∈|² - 1)^u∈ = 0 for u^∈ : R^d → R², as well as some natural quasilinear generalizations for functions taking values in R^k, k ≥ 2. We prove that for solutions of the general system, the limiting support as ∈ → 0 of the energy measure is a codimension k manifold which evolves via mean curvature. We also establish some local regularity results which hold uniformly in ∈. In particular, we establish a small-energy regulity theorem for the general system, and we prove a stronger regularity result for the usual Ginzburg-Landau equation on R².