Quantitative convergence of the vectorial AllenCahn equation towards multiphase mean curvature flow
Abstract
Phasefield models such as the AllenCahn 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 sharpinterface limit, the vectorial AllenCahn equation with a potential with $N\geq 3$ 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 AllenCahn equation with wellprepared initial data converge towards multiphase mean curvature flow in the limit of vanishing interface width parameter $\varepsilon\searrow 0$. We even establish the rate of convergence $O(\varepsilon^{1/2})$. Our approach is based on the gradient flow structure of the AllenCahn 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 AllenCahn equation with multiwell potential. This enables us to overcome the limitations of other approaches, e.g. avoiding the need for a stability analysis of the AllenCahn operator or additional convergence hypotheses for the energy at positive times.
 Publication:

arXiv eprints
 Pub Date:
 March 2022
 DOI:
 10.48550/arXiv.2203.17143
 arXiv:
 arXiv:2203.17143
 Bibcode:
 2022arXiv220317143F
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Numerical Analysis
 EPrint:
 53 pages