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

### Julian Fischer

Institute of Science and Technology Austria (IST Austria), Klosterneuburg, Austria### Alice Marveggio

Institute of Science and Technology Austria (IST Austria), Klosterneuburg, Austria

## Abstract

Phase-field models such as the Allen–Cahn equation may give rise to the formation and evolution of geometric shapes, a phenomenon that may be analyzed rigorously in suitable scaling regimes. In its sharp-interface limit, the vectorial Allen–Cahn equation with a potential with $N≥3$ distinct minima has been conjectured to describe the evolution of branched interfaces by multiphase mean curvature flow. In the present work, we give a rigorous proof for this statement in two and three ambient dimensions and for a suitable class of potentials: as long as a strong solution to multiphase mean curvature flow exists, solutions to the vectorial Allen–Cahn equation with well-prepared initial data converge towards multiphase mean curvature flow in the limit of vanishing interface width parameter $ε↘0$. We even establish the rate of convergence $O(ε_{1/2})$. Our approach is based on the gradient-flow structure of the Allen–Cahn equation and its limiting motion: building on the recent concept of “gradient-flow calibrations” for multiphase mean curvature flow, we introduce a notion of relative entropy for the vectorial Allen–Cahn equation with multi-well potential. This enables us to overcome the limitations of other approaches, e.g. avoiding the need for a stability analysis of the Allen–Cahn operator or additional convergence hypotheses for the energy at positive times.

## Cite this article

Julian Fischer, Alice Marveggio, Quantitative convergence of the vectorial Allen–Cahn equation towards multiphase mean curvature flow. Ann. Inst. H. Poincaré C Anal. Non Linéaire (2024), published online first

DOI 10.4171/AIHPC/109