ANALYSIS OF A DIFFUSE INTERFACE APPROACH TO AN ADVECTION DIFFUSION EQUATION ON A MOVING SURFACE
Charles M. Elliott, Björn Stinner · Mathematical Models and Methods in Applied Sciences · 2009
A diffuse interface model for an advection diffusion equation on a moving surface is formulated involving a small parameter ε related to the thickness of the interfacial layer. The coefficient functions degenerate on the boundary of the diffuse interface. In appropriately weighted Sobolev spaces, existence and uniqueness of weak solutions is shown. Using energy methods the convergence of solutions to the diffuse interface model to the solution to the equation on the moving surface as ε → 0 is proved. The approach is intended to be applied to phase field models describing the surface motion. Among other problems we have surfactants on liquid-liquid interfaces and species diffusion on moving grain boundaries in mind.