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∈ : Rd → R2, as well as some natural quasilinear generalizations for functions taking values in Rk, 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.
|Original language||English (US)|
|Number of pages||44|
|Journal||Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire|
|State||Published - 1999|
All Science Journal Classification (ASJC) codes
- Mathematical Physics
- Applied Mathematics