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Phase-field models such as the Allen-Cahn equation may give rise to theformation and evolution of geometric shapes, a phenomenon that may be analyzedrigorously in suitable scaling regimes. In its sharp-interface limit, thevectorial Allen-Cahn equation with a potential with $N\geq 3$ distinct minimahas been conjectured to describe the evolution of branched interfaces bymultiphase mean curvature flow.  In the present work, we give a rigorous proof for this statement in two andthree ambient dimensions and for a suitable class of potentials: As long as astrong solution to multiphase mean curvature flow exists, solutions to thevectorial Allen-Cahn equation with well-prepared initial data converge towardsmultiphase mean curvature flow in the limit of vanishing interface widthparameter $\varepsilon\searrow 0$. We even establish the rate of convergence$O(\varepsilon^{1/2})$.  Our approach is based on the gradient flow structure of the Allen-Cahnequation and its limiting motion: Building on the recent concept of "gradientflow calibrations" for multiphase mean curvature flow, we introduce a notion ofrelative entropy for the vectorial Allen-Cahn equation with multi-wellpotential. This enables us to overcome the limitations of other approaches,e.g. avoiding the need for a stability analysis of the Allen-Cahn operator oradditional convergence hypotheses for the energy at positive times.

Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow