Nonlocal Phase Field Models of Viscous Cahn–Hilliard Type

A nonlocal phase field model of viscous Cahn–Hilliard type is considered. This model constitutes a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion that has been studied in a series of papers by P. Podio-Guidugli and the present authors. The resulting system of differential equations consists of a highly nonlinear parabolic equation coupled to a nonlocal ordinary differential equation, which has singular terms that render the analysis difficult. Some results are presented on the well-posedness and stability of the system as well as on the distributed optimal control problem.