Asymptotic and numerical analysis of the Allen-Cahn equation with a mass constraint

• Main Author: Stafford, Douglas James
• Language: English
• Published: 2009
• Online Access: http://hdl.handle.net/2429/6432

The Allen-Cahn equation with a mass constraint is analyzed asymptotically and numerically in a two-dimensional domain. This problem models the phase separation of a binary mixture in the presence of a mass constraint. Solutions develop internal layers, or interfaces, that propagate depending on the curvature of the interfaces while keeping the area they enclose constant. Small interfaces attached to the boundary of the domain are shown to move along the boundary in the direction of increasing boundary curvature. The motion of the interfaces is simulated numerically to verify these asymptotic results. The slow motion behavior of a semi-circular interface intersecting aflat boundary segment is also analyzed. The projection method is used to derive an explicit ordinary differential equation for the location of the center of such a semi-circular interface.