Analysis of models for curvature driven motion of interfaces

Swartz, Drew E. PhD, Purdue University, December 2015. Analysis of Models for Curvature Driven Motion of Interfaces. Major Professor: Nung Kwan Yip. Interfacial energies frequently appear in models arising in materials science and engineering. To dissipate energy in these systems, the interfaces will often move by a curvature dependent velocity. The present work details the mathematical analysis of some models for curvature dependent motion of interfaces. In particular we focus on two types, thresholding schemes and phase field models. With regard to thresholding schemes, we give a new proof of the convergence of the Merriman-Bence-Osher thresholding algorithm to motion by mean curvature. This new proof does not rely on the scheme satisfying a comparison principle. The technique shows promise in proving the convergence of thresholding schemes for more general motions, such as fourth-order motions and motions of higher codimension interfaces. The application of the proof technique to these more general schemes is discussed, along with rigorous consistency estimates. With regard to phase-field models, we examine the L-gradient flow of a second order gradient model for phase transitions, introduced by Fonseca and Mantegazza. In the case of radial symmetry we demonstrate that the diffuse interfacial dynamics converge to motion by mean curvature as the width of the interface decreases to zero. This is in accordance with the first-order Allen-Cahn model for phase transitions. But unlike the Allen-Cahn model, the gradient flow for the Fonseca-Mantegazza model is a fourth-order parabolic PDE. This creates new and novel difficulties in its analysis.