@article{01310cdaf96e4f048e87fc3af3e937a4,

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 = "Funding Information: \ast Received by the editors February 27, 2020; accepted for publication (in revised form) August 6, 2020; published electronically December 15, 2020. https://doi.org/10.1137/20M1322182 Funding: This work was supported by the European Union's Horizon 2020 Research and Innovation Programme under Marie Sk\lodowska-Curie grant agreement 665385 and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy, EXC-2047/1--390685813. \dagger IST Austria, Klosterneuburg 3400, Austria (julian.fischer@ist.ac.at). \ddagger Hausdorff Center for Mathematics, University of Bonn, Bonn 53115, Germany (tim.laux@hcm. uni-bonn.de). \S Institute for Applied Mathematics, University of Bonn, 53115 Bonn, Germany (simon@iam.uni-bonn.de). Publisher Copyright: {\textcopyright} 2020 SIAM. Published by SIAM under the terms of the Creative Commons 4.0 license",

year = "2020",

month = dec,

day = "15",

doi = "10.1137/20M1322182",

language = "English (US)",

volume = "52",

pages = "6222--6233",

journal = "SIAM Journal on Mathematical Analysis",

issn = "0036-1410",

publisher = "Society for Industrial and Applied Mathematics Publications",

number = "6",

}