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

Fischer, J.; Laux, T.; Simon, T.M.

Research article (journal) | Peer reviewed

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.

Details about the publication

JournalSIAM Journal on Mathematical Analysis (SIAM J. Math. Anal.)
Volume52
Issue6
Page range6222-6233
StatusPublished
Release year2020
Language in which the publication is writtenEnglish
DOI10.1137/20M1322182
Keywordsmean curvature flow; Allen--Cahn equation; relative entropy method; diffuse interface; reaction-diffusion equations

Authors from the University of Münster

Simon, Theresa
Juniorprofessorship of Applied Mathematics (Prof. Simon)