DYNAMIC TRANSITIONS AND PATTERN FORMATIONS FOR CAHN-HILLIARD MODEL WITH LONG-RANGE REPULSIVE INTERACTIONS

Abstract. The main objective of this article is to study the order-disorder phase transition and pattern formation for systems with long-range repulsive interactions. The main focus is on the Cahn-Hilliard model with a nonlocal term in the corresponding energy functional, representing the long-range re-pulsive interaction. First, we show that as soon as the linear problem loses stability, the system always undergoes a dynamic transition to one of the three types, forming different patterns/structures. The types of transition are then dictated by a nondimensional parameter, measuring the interactions between the long-range repulsive term and the quadratic and cubic nonlinearities in the model. The derived explicit form of this parameter offers precise informa-tion for the phase diagrams. Second, we obtain a novel and explicit pattern selection mechanism associated with the competition between the long-range repulsive interaction and the short-range attractive interactions. In particular, the hexagonal pattern is unique to the long-range interaction, and is associ-ated with a novel two-dimensional reduced transition equations on the cente