Kilian Koch

Publications

• Thresholding Scheme for the Half-Harmonic Map Heat Flow
  arXiv (2026-09-22) preprint
  By Kilian Koch, Christof Melcher, Endre Süli
  Read Online DOI
  Show Abstract

  We propose a thresholding scheme for the half-harmonic map heat flow from the flat torus $\mathbb{T}^d$ into the unit sphere $\mathbb{S}^m$ in $\mathbb{R}^{m+1}$, 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.
• Well-posedness of half-harmonic map heat flows for rough initial data
  arXiv (2025-04-09) preprint
  By Kilian Koch, Christof Melcher
  Read Online DOI
  Show Abstract

  We adopt the Koch-Tataru theory for the Navier-Stokes equations, based on Carleson measure estimates, to develop a scaling-critical low-regularity framework for half-harmonic map heat flows. This nonlocal variant of the harmonic map heat flow has been studied recently in connection with free boundary minimal surfaces. We introduce a new class of initial data for the flow, broader than the conventional energy or Sobolev spaces considered in previous work, for which we establish existence, uniqueness, and continuous dependence. The class particularly includes homogeneous initial data that give rise to self-similar expanders.

Talks

• 
  Thresholding Scheme for the Half-Harmonic Map Heat Flow

  Young Researchers Workshop on Geometric Analysis, Singular PDEs and Numerics, RWTH Aachen University · 1 June 2026

  Slides (PDF) Event page

  Show Abstract

  We study the half-harmonic map heat flow, the gradient flow of the half-Dirichlet energy for sphere-valued maps on the torus. Motivated by the thresholding algorithm of Laux and Yip for mean curvature flow, we replace the classical Gaussian kernel by the Poisson kernel, the semigroup kernel associated to the half-Laplacian, and alternate convolution with pointwise projection onto the sphere. We prove that the resulting iterates converge, along a subsequence, to a weak solution of the flow attaining the initial data in H^1/2 and satisfying an energy inequality. We also consider a truncated variant of the scheme, where the Poisson kernel is projected onto the first N Fourier modes. Here convergence holds under the condition that hN − d(d+1) log(1/h) → ∞, meaning N must grow slightly faster than log(1/h)/h as h → 0. This truncated version is intended as a first step toward a fully implementable numerical method. As a complement, we establish weak-strong uniqueness: a weak solution satisfying the energy inequality must coincide with any strong solution sharing the same initial data, showing in particular that the scheme converges to the strong solution whenever one exists.
• 
  Well-posedness of half-harmonic map heat flows for rough initial data

  3 City Seminar 2025 (invited talk), TU Eindhoven, Netherlands · 1 October 2025

  Slides (PDF)

  Show Abstract

  We adopt the Koch–Tataru theory for the Navier–Stokes equations, based on Carleson measure estimates, to develop a scaling-critical low-regularity framework for half-harmonic map heat flows. This nonlocal variant of the harmonic map heat flow has been studied recently in connection with free boundary minimal surfaces. We introduce a new class of initial data for the flow, broader than the conventional energy or Sobolev spaces considered in previous work, for which we establish existence, uniqueness, and continuous dependence. The class particularly includes homogeneous initial data that give rise to self-similar expanders.

Organized Events

• Young Researchers Workshop on Geometric Analysis, Singular PDEs and Numerics Co-organizer

  Joint workshop of CRC 1481, EDDy and IntComSin (FAU Erlangen-Nürnberg / University of Regensburg), RWTH Aachen University · 1 June 2026 – 3 June 2026

  Event page

Research Project & Funding

Project C01: Singularity formation in dissipative harmonic flows

My position is funded by the German Research Foundation (DFG) through CRC 1481 Sparsity and Singular Structures (project no. 442047500), project C01, since October 2024.

Project page ↗  |  CRC 1481 ↗