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Interacting fronts in the strongly nonlocal Allen--Cahn equation

  • Erisa Hasani
  • Stefania Patrizi
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Abstract

We study the sharp-interface limit of the fractional Allen--Cahn equation in Rn, n≥ 2, in the strongly nonlocal regime s∈(0,1/2), for initial data consisting of finitely many nested transition layers. We identify a coupled geometric law for the motion of the resulting fronts: each velocity contains fractional mean curvature and a nonlocal interaction potential generated by the other fronts. These interactions persist while the fronts remain separated, in contrast to the independent motion in the critical regime s=1/2. Assuming that the coupled law admits a smooth evolution with strictly nested sets on a given time interval, we prove that the Allen--Cahn solutions converge locally uniformly away from the moving fronts to the corresponding integer-valued phases.

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Cite

BibTeXRIS
BibTeX
@misc{hasani2026interacting,
  title = {{Interacting fronts in the strongly nonlocal Allen--Cahn equation}},
  author = {Erisa Hasani and Stefania Patrizi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.35666},
  eprint = {2609.35666},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.35666},
}
RIS
TY  - UNPB
TI  - Interacting fronts in the strongly nonlocal Allen--Cahn equation
AU  - Erisa Hasani
AU  - Stefania Patrizi
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.35666
UR  - https://arxiv.org/abs/2609.35666
AB  - We study the sharp-interface limit of the fractional Allen--Cahn equation in R^n, n≥ 2, in the strongly nonlocal regime s∈(0,1/2), for initial data consisting of finitely many nested transition layers. We identify a coupled geometric law for the motion of the resulting fronts: each velocity contains fractional mean curvature and a nonlocal interaction potential generated by the other fronts. These interactions persist while the fronts remain separated, in contrast to the independent motion in the critical regime s=1/2. Assuming that the coupled law admits a smooth evolution with strictly nested sets on a given time interval, we prove that the Allen--Cahn solutions converge locally uniformly away from the moving fronts to the corresponding integer-valued phases.
ER  -