Kilian Koch
数学・博士課程研究員
アーヘン工科大学 応用解析講座
幾何解析における勾配流、特に(半)調和写像熱流の適切性と特異性に ついて研究しています。アーヘン工科大学のCRC 1481 Sparsity and Singular Structures の博士課程研究員です。
研究について詳しく
幾何解析における勾配流、特に(半)調和写像熱流の適切性と特異性に ついて研究しています。アーヘン工科大学のCRC 1481 Sparsity and Singular Structures の博士課程研究員です。
0009-0008-4358-9245 arXiv GitHub LinkedIn koch@math1.rwth-aachen.de
論文
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Well-posedness of half-harmonic map heat flows for rough initial dataarXiv (2025-04-09) プレプリント
概要を表示
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.
講演
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Thresholding Scheme for the Half-Harmonic Map Heat FlowYoung Researchers Workshop on Geometric Analysis, Singular PDEs and Numerics, RWTH Aachen University · 2026年6月1日スライド (PDF) イベントページ概要を表示
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 data3 City Seminar 2025 (invited talk), TU Eindhoven, Netherlands · 2025年10月1日スライド (PDF)概要を表示
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.
主催イベント
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Young Researchers Workshop on Geometric Analysis, Singular PDEs and Numerics Co-organizerJoint workshop of CRC 1481, EDDy and IntComSin (FAU Erlangen-Nürnberg / University of Regensburg), RWTH Aachen University · 2026年6月1日 – 2026年6月3日イベントページ
研究プロジェクト・資金
プロジェクトC01: Singularity formation in dissipative harmonic flows
私のポジションは、2024年4月よりドイツ研究振興協会(DFG)の CRC 1481 Sparsity and Singular Structures (プロジェクト番号 442047500)、プロジェクトC01の資金提供を 受けています。