Asymptotics for two-dimensional parabolic vectorial Allen-Cahn systems
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.
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BibTeX
@misc{dai2026asymptotics,
title = {{Asymptotics for two-dimensional parabolic vectorial Allen-Cahn systems}},
author = {Zhiyuan Dai and Haotong Fu and Huaijie Wang and Wei Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27596},
eprint = {2609.27596},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27596},
}
RIS
TY - UNPB
TI - Asymptotics for two-dimensional parabolic vectorial Allen-Cahn systems
AU - Zhiyuan Dai
AU - Haotong Fu
AU - Huaijie Wang
AU - Wei Wang
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27596
UR - https://arxiv.org/abs/2609.27596
AB - 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.
ER -