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

Fischer JL, Laux T, Simon TM. 2020. Convergence rates of the Allen-Cahn equation to mean curvature flow: A short proof based on relative entropies. SIAM Journal on Mathematical Analysis. 52(6), 6222–6233.

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Journal Article | Published | English

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Author
Fischer, Julian LIST Austria ; Laux, Tim; Simon, Theresa M.
Department
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.
Publishing Year
Date Published
2020-12-15
Journal Title
SIAM Journal on Mathematical Analysis
Acknowledgement
This work was supported by the European Union's Horizon 2020 Research and Innovation Programme under Marie Sklodowska-Curie grant agreement 665385 and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy, EXC-2047/1--390685813.
Volume
52
Issue
6
Page
6222-6233
ISSN
eISSN
IST-REx-ID

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Fischer JL, Laux T, Simon TM. Convergence rates of the Allen-Cahn equation to mean curvature flow: A short proof based on relative entropies. SIAM Journal on Mathematical Analysis. 2020;52(6):6222-6233. doi:10.1137/20M1322182
Fischer, J. L., Laux, T., & Simon, T. M. (2020). Convergence rates of the Allen-Cahn equation to mean curvature flow: A short proof based on relative entropies. SIAM Journal on Mathematical Analysis. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/20M1322182
Fischer, Julian L, Tim Laux, and Theresa M. Simon. “Convergence Rates of the Allen-Cahn Equation to Mean Curvature Flow: A Short Proof Based on Relative Entropies.” SIAM Journal on Mathematical Analysis. Society for Industrial and Applied Mathematics, 2020. https://doi.org/10.1137/20M1322182.
J. L. Fischer, T. Laux, and T. M. Simon, “Convergence rates of the Allen-Cahn equation to mean curvature flow: A short proof based on relative entropies,” SIAM Journal on Mathematical Analysis, vol. 52, no. 6. Society for Industrial and Applied Mathematics, pp. 6222–6233, 2020.
Fischer JL, Laux T, Simon TM. 2020. Convergence rates of the Allen-Cahn equation to mean curvature flow: A short proof based on relative entropies. SIAM Journal on Mathematical Analysis. 52(6), 6222–6233.
Fischer, Julian L., et al. “Convergence Rates of the Allen-Cahn Equation to Mean Curvature Flow: A Short Proof Based on Relative Entropies.” SIAM Journal on Mathematical Analysis, vol. 52, no. 6, Society for Industrial and Applied Mathematics, 2020, pp. 6222–33, doi:10.1137/20M1322182.
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