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We prove a quantitative convergence result of the nonlocal Allen--Cahnequation to volume-preserving mean curvature flow. The proof uses gradient flowcalibrations and the relative entropy method, which has been used in the recentliterature to prove weak-strong uniqueness results for mean curvature flow andconvergence of the Allen--Cahn equation. A crucial difference in this work is anew notion of gradient flow calibrations. We add a tangential component to thevelocity field in order to prove the Gronwall estimate for the relative energy.This allows us to derive the optimal convergence rate without having to showthe closeness of the Lagrange-multipliers.

Quantitative convergence of the nonlocal Allen--Cahn equation to volume-preserving mean curvature flow