发表机构
RWTH Aachen University; University of Oxford（亚琛工业大学; 牛津大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种基于泊松核卷积与逐点归一化的阈值化格式求解半调和映射热流，证明其收敛到满足能量耗散的弱解，并建立了弱-强唯一性。
AI 中文摘要
我们针对从平坦环面 $\mathbb{T}^d$ 到 $\mathbb{R}^{m+1}$ 中单位球面 $\mathbb{S}^m$ 的半调和映射热流提出了一种阈值化格式，其迭代由与泊松核的卷积给出，取代了经典 Merriman-Bence-Osher 算法中的高斯核，并进行逐点归一化。我们证明了分段常数插值子收敛到半调和映射热流的一个弱解，该弱解在能量空间中强达到初始映射，并满足能量耗散不等式。对于谱截断变体，在步长满足一定条件时收敛成立。一个关键要素是直接从精细的步内估计推导出精确的能量耗散恒等式，而无需借助最小化移动框架中通常使用的 De Giorgi 插值。这提供了一种直接的方式在极限中恢复适当的能量耗散性质，进而实现弱-强唯一性：只要强解存在，每个能量耗散的弱解都与强解一致。
英文摘要
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.
