Thresholding Scheme for the Half-Harmonic Map Heat Flow

Abstract

We propose a thresholding scheme for the half-harmonic map heat flow from the flat torus 𝕋^d into the unit sphere 𝕊^m in ℝ^m+1, whose iterates are given by convolution with the Poisson kernel, replacing the Gaussian kernel of the classical Merriman-Bence-Osher algorithm, and pointwise normalization. We prove that the piecewise constant interpolants subconverge to a weak solution of the half-harmonic maps heat flow that attains the initial map strongly in the energy space and satisfies the energy-dissipation inequality. For a spectrally truncated variant, convergence holds under a condition on the step size. A key ingredient is the derivation of the precise energy-dissipation identity directly from a refined within-step estimate, without resorting to De Giorgi interpolation typically used in the minimizing movement framework. This provides a direct way to recover the appropriate energy-dissipation property in the limit and, in turn, enables weak-strong uniqueness: every energy-dissipating weak solution coincides with the strong solution whenever the latter exists.

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BibTeX RIS

BibTeX

@misc{koch2026thresholding,
  title = {{Thresholding Scheme for the Half-Harmonic Map Heat Flow}},
  author = {Kilian Koch and Christof Melcher and Endre Süli},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.25912},
  eprint = {2609.25912},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.25912},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Thresholding Scheme for the Half-Harmonic Map Heat Flow
AU  - Kilian Koch
AU  - Christof Melcher
AU  - Endre Süli
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.25912
UR  - https://arxiv.org/abs/2609.25912
AB  - We propose a thresholding scheme for the half-harmonic map heat flow from the flat torus 𝕋^d into the unit sphere 𝕊^m in ℝ^m+1, whose iterates are given by convolution with the Poisson kernel, replacing the Gaussian kernel of the classical Merriman-Bence-Osher algorithm, and pointwise normalization. We prove that the piecewise constant interpolants subconverge to a weak solution of the half-harmonic maps heat flow that attains the initial map strongly in the energy space and satisfies the energy-dissipation inequality. For a spectrally truncated variant, convergence holds under a condition on the step size. A key ingredient is the derivation of the precise energy-dissipation identity directly from a refined within-step estimate, without resorting to De Giorgi interpolation typically used in the minimizing movement framework. This provides a direct way to recover the appropriate energy-dissipation property in the limit and, in turn, enables weak-strong uniqueness: every energy-dissipating weak solution coincides with the strong solution whenever the latter exists.
ER  -

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