Colliding Interfaces in Old and New Diffuse-interface Approximations of Willmore-flow

TL;DR

This paper investigates diffuse-interface models for Willmore flow, highlighting issues with standard models during interface collisions and proposing alternative models that better capture the true sharp-interface dynamics.

Contribution

It introduces new diffuse-interface approximations based on energies converging to the lower-semicontinuous envelope of Willmore energy, improving collision behavior.

Findings

Standard models can produce non-physical corners during collisions.

Proposed models lead to more regular interface evolution.

Alternative energies better approximate the sharp-interface limit.

Abstract

This paper is concerned with diffuse-interface approximations of the Willmore flow. We first present numerical results of standard diffuse-interface models for colliding one dimensional interfaces. In such a scenario evolutions towards interfaces with corners can occur that do not necessarily describe the adequate sharp-interface dynamics. We therefore propose and investigate alternative diffuse-interface approximations that lead to a different and more regular behavior if interfaces collide. These dynamics are derived from approximate energies that converge to the $L^1$-lower-semicontinuous envelope of the Willmore energy, which is in general not true for the more standard Willmore approximation.