General regularity obstructions for the arrival time equation

Abstract

We study the regularity of the arrival time function associated with mean curvature flow, formulated as a degenerate elliptic equation encoding the singular structure of the flow. We identify a systematic mechanism obstructing higher regularity, arising from higher-order asymptotic expansions near spherical singularities. We construct uncountably many low-regularity arrival time functions in all dimensions. In particular, we settle the question of smoothness in the planar case by showing that even for convex curve shortening flow the arrival time need not be C²⁵. Our companion paper proves that every such arrival time is C^24,α for any 0<α<1. Hence the obstruction at order $25$ gives the optimal regularity threshold. Our approach introduces new analytic ingredients, including a precise correspondence between elliptic asymptotic expansions and parabolic long-time asymptotics of the associated rescaled mean curvature flow, a complexification of the arrival time equation, and a method for prescribing higher-order asymptotic expansions.

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BibTeX

@misc{huang2026general,
  title = {{General regularity obstructions for the arrival time equation}},
  author = {Yiqi Huang and Jingze Zhu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.31599},
  eprint = {2609.31599},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.31599},
}

RIS

TY  - UNPB
TI  - General regularity obstructions for the arrival time equation
AU  - Yiqi Huang
AU  - Jingze Zhu
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.31599
UR  - https://arxiv.org/abs/2609.31599
AB  - We study the regularity of the arrival time function associated with mean curvature flow, formulated as a degenerate elliptic equation encoding the singular structure of the flow. We identify a systematic mechanism obstructing higher regularity, arising from higher-order asymptotic expansions near spherical singularities. We construct uncountably many low-regularity arrival time functions in all dimensions. In particular, we settle the question of smoothness in the planar case by showing that even for convex curve shortening flow the arrival time need not be C^25. Our companion paper proves that every such arrival time is C^24,α for any 0<α<1. Hence the obstruction at order $25$ gives the optimal regularity threshold. Our approach introduces new analytic ingredients, including a precise correspondence between elliptic asymptotic expansions and parabolic long-time asymptotics of the associated rescaled mean curvature flow, a complexification of the arrival time equation, and a method for prescribing higher-order asymptotic expansions.
ER  -

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