Dynamics of Phase Separation and Pattern Formation

The focus of this thesis is the study of the evolution of two models adopted in the context of phase separation and pattern formation, the Cahn-Hilliard model and the Swift-Hohenberg model. In the study of the Cahn-Hilliard model, the PDEs arising as the L² and H^-1 gradient flows in the higher dimensional setting n > 1 are studied, and estimates are provided on the evolution of solutions initiated close to configurations that globally or locally minimize the perimeter of the interface are provided. The results rely on a new regularity property of a local version of the well-known isoperimetric function. In the Swift-Hohenberg setting, the one dimensional model is considered, and the slow evolution of a particular class of solutions is established. In this context, existence and regularity of solutions in dimension n_< 3 are provided. In the last part of this thesis, two ongoing project and future research directions are presented.