Thresholding Scheme for the Half-Harmonic Map Heat Flow

arXiv (Cornell University)

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.

Authors 3

1. Kilian Koch Aachen Lehrstuhl für Angewandte Analysis

   RWTH Aachen University

   Affiliation as printed

   Lehrstuhl für Angewandte Analysis , RWTH Aachen University , Kreuzherren- str. 2 , 52062 Aachen , Germany

2. Christof Melcher Aachen Lehrstuhl für Angewandte Analysis

   RWTH Aachen University

   Affiliation as printed

   Lehrstuhl für Angewandte Analysis , RWTH Aachen University , Kreuzherrenstr. 2 , 52062 Aachen , Ger- many

3. Endre Süli

   University of Oxford

   Affiliation as printed

   Mathematical Institute , University of Oxford , Andrew Wiles Building , Radcliffe Observatory Quarter , Woodstock Road , Oxford OX2 6GG , United Kingdom