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We give a short and self-contained proof for rates of convergence of theAllen-Cahn equation towards mean curvature flow, assuming that a classical(smooth) solution to the latter exists and starting from well-prepared initialdata. Our approach is based on a relative entropy technique. In particular, itdoes 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 tobe applicable to more complex equations and systems.

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