Gradient Flows of Interfacial Energies: Curvature Agents and Incompressibility

A framework for the gradient flow of sharp-interface surface energies that couple to embedded curvature-active agents is developed for one-dimensional curves embedded in [Formula: see text]. A [Formula: see text]-limit is derived for an Allen–Cahn-type model that couples surface agents to curvature energy. A penalty method that maintains spatially uniform interfacial mass is introduced; formal multiscale analysis is used to derive a corresponding locally incompressible gradient flow that couples interface stretching or compression to local flux of interfacial mass. This is supported by a rigorous analysis of the convergence of the penalty method to an incompressible limiting system for a narrower class of surface energies and initial data. Computational results are presented in support of the analysis.