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

Julian Fischer; Tim Laux; Theresa M. Simon

doi:10.1137/20M1322182

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

Julian Fischer, Tim Laux, Theresa M. Simon

Research output: Contribution to journal › Article › peer-review

28 Scopus citations

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.

Original language	English (US)
Pages (from-to)	6222-6233
Number of pages	12
Journal	SIAM Journal on Mathematical Analysis
Volume	52
Issue number	6
DOIs	https://doi.org/10.1137/20M1322182
State	Published - Dec 15 2020
Externally published	Yes

All Science Journal Classification (ASJC) codes

Analysis
Computational Mathematics
Applied Mathematics

Keywords

Allen-Cahn equation
Diffuse interface
Mean curvature flow
Reaction-diffusion equations
Relative entropy method

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10.1137/20M1322182

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title = "Convergence rates of the allen-cahn equation to mean curvature flow: A short proof based on relative entropies",

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.",

keywords = "Allen-Cahn equation, Diffuse interface, Mean curvature flow, Reaction-diffusion equations, Relative entropy method",

author = "Julian Fischer and Tim Laux and Simon, \{Theresa M.\}",

note = "Publisher Copyright: {\textcopyright} 2020 SIAM. Published by SIAM under the terms of the Creative Commons 4.0 license",

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T1 - Convergence rates of the allen-cahn equation to mean curvature flow

T2 - A short proof based on relative entropies

AU - Fischer, Julian

AU - Laux, Tim

AU - Simon, Theresa M.

PY - 2020/12/15

Y1 - 2020/12/15

N2 - 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.

AB - 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.

KW - Allen-Cahn equation

KW - Diffuse interface

KW - Mean curvature flow

KW - Reaction-diffusion equations

KW - Relative entropy method

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SP - 6222

EP - 6233

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

IS - 6

ER -

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

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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