# 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 Anal. Non Linéaire 41 (2024), no. 5, pp. 1117–1178

DOI 10.4171/AIHPC/109