| The Cahn-Hilliard equation is a fourth-order parabolic partial differential equation which is one of the leading models for the study of phase separation in isothermal, isotropic mixtures. The goal of this dissertation is to provide insight into the qualitative dynamic properties of solutions of the one-dimensional Cahn-Hilliard equation. The main focus is on the early stages of evolution of solutions whose initial data is nearly uniform. Linear and numerical analysis has led to the conjecture of the existence of a large class of such solutions that evolve relatively quickly to become nearly periodic with large amplitude and small period. Such solutions would correspond to the experimentally-observed phenomenon of spinodal decomposition, the fine-grained decomposition of a molten binary alloy after it has been rapidly quenched. In this dissertation, I present a rigorous mathematical justification for the process of spinodal decomposition. I believe that this is the first rigorous treatment of this phenomenon.; As a complement to these precise results, the last part of this dissertation deals with certain formal and asymptotic methods for studying aspects of the Cahn-Hilliard equation which have not yet been settled by rigorous techniques. In particular, models which approximate the Cahn-Hilliard equation with a finite system of ordinary differential equations are derived in various contexts, and their behavior is described. |
