Convergence rates of the Allen-Cahn equation to mean curvature flow: A short proof based on relative entropies

2020/02/27 by Julian Fischer, Tim Laux, Fischer, Julian +3 · 5 citations
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Meromorphic and Entire Functions #Nonlinear Partial Differential Equations #math.AP

paper · pdf · doi:10.48550/arxiv.2002.11994

12 pages

openalex publication_date 2020/02/27 · arxiv created 2020/08/18 · arxiv updated 2020/08/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give a short and self-contained proof for rates of convergence of the Allen-Cahn equation towards mean curvature flow, assuming that a classical (smooth) solution to the latter exists and starting from well-prepared initial data. Our approach is based on a relative entropy technique. In particular, it does not require a stability analysis for the linearized Allen-Cahn operator. As our analysis also does not rely on the comparison principle, we expect it to be applicable to more complex equations and systems.

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