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Convergence Rates of the Allen--Cahn Equation to Mean Curvature Flow: A Short Proof Based on Relative Entropies
SIAM Journal on Mathematical Analysis ( IF 2.2 ) Pub Date : 2020-12-15 , DOI: 10.1137/20m1322182
Julian Fischer , Tim Laux , Theresa M. Simon

SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 6222-6233, January 2020.
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.


中文翻译:

Allen-Cahn方程对平均曲率流的收敛速度:基于相对熵的简短证明

SIAM数学分析杂志,第52卷,第6期,第6222-6233页,2020年1月。
我们给出了关于Allen-Cahn方程向平均曲率流的收敛速度的简短且独立的证明,假设经典(平滑的解决方案存在,并且要从准备充分的初始数据开始。我们的方法基于相对熵技术。特别是,它不需要线性化Allen-Cahn算子的稳定性分析。由于我们的分析也不依赖比较原理,因此我们希望它适用于更复杂的方程式和系统。
更新日期:2020-12-16
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