A phase-field approximation of the Willmore flow with volume constraint

TL;DR

This paper establishes the well-posedness of a phase-field model approximating the Willmore flow with a volume constraint, using a variational approach based on the gradient flow structure.

Contribution

It introduces a novel phase-field approximation for the constrained Willmore flow and proves its well-posedness through a variational minimization framework.

Findings

Existence of solutions for the phase-field approximation

Utilization of gradient flow structure for proof

Variational minimization approach for time discretization

Abstract

The well-posedness of a phase-field approximation to the Willmore flow with volume constraint is established. The existence proof relies on the underlying gradient flow structure of the problem: the time discrete approximation is solved by a variational minimization principle.