Colliding Interfaces in Old and New Diffuse-interface Approximations of Willmore-flow
For researchers in geometric flows and interface modeling, this work addresses a known limitation of diffuse-interface approximations, but the contribution is incremental as it focuses on a specific scenario (1D collisions) without broad validation.
The paper identifies that standard diffuse-interface models for Willmore flow produce corner-like singularities during interface collisions, deviating from sharp-interface dynamics. The authors propose alternative approximations that yield more regular behavior, derived from energies converging to the L1-lower-semicontinuous envelope of the Willmore energy.
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.