Year: 2026
Type: preprint
Abstract: We study the parabolic vectorial Allen--Cahn equation in two space dimensions for a fixed smooth potential with finitely many non-degenerate wells. Our result provides a parabolic extension of Bethuel's elliptic compactness theorem: at almost every positive time, the diffuse energy, potential energy, and gradient tensor converge to measures concentrated on a countably \(1\)-rectifiable interface and satisfy all the Bethuel-type relations. In the scalar case, the tangential defect vanishes and the usual mean-curvature-flow structure is recovered. For general vectorial potentials, the limiting stress and dissipation yield a two-mobility system: the normal energy flux balances weighted curvature, while a tangential internal mobility transports residual tangential energy. Thus the result extends the Allen--Cahn-to-Brakke framework of Ilmanen from scalar to vector-valued systems, with an additional non-negative dissipation defect in the localized energy inequality.
Source: arXiv (Cornell University)
Authors: Dai, Zhiyuan, Fu, Haotong, Wang, Huaijie, Wang, Wei
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214237569