A Cahn–Hilliard–Willmore phase field model for non-oriented interfaces

Abstract

We investigate a new phase field model for representing non-oriented interfaces, approximating their area and simulating their area-minimizing flow. Our contribution is related to the approach proposed by Bretin et al. (2022) that involves ad hoc neural networks. We show here that, instead of neural networks, similar results can be obtained using a more standard variational approach that combines a Cahn–Hilliard-type functional involving an appropriate non-smooth potential and a Willmore-type stabilization energy. We give a $\Gamma$-convergence analysis of this phase field model in dimension $1$ and, for radially symmetric functions, in arbitrary dimension. We also propose a simple numerical scheme to approximate its $L^2$-gradient flow. We illustrate numerically that the new flow approximates fairly well the mean curvature flow of codimension $1$ or $2$ interfaces in dimensions $2$ and $3$.