### Abstract

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} → R^{2}, 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 R^{2}.

Original language | English (US) |
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Pages (from-to) | 423-466 |

Number of pages | 44 |

Journal | Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire |

Volume | 16 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 1999 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Analysis
- Mathematical Physics
- Applied Mathematics