Spinodal Decomposition for the Cahn-hilliard Equation

The Cahn-Hilliard equation is a fourth-order parabolic partial differential equation that is one of the leading models for the study of phase separation in isothermal, isotropic, binary mixtures, such as molten alloys. When a spatially homogeneous alloy is rapidly quenched in a physical experiment, a fine-grained decomposition into two distinct phases is frequently observed; this phenomenon is known as spinodal decomposition. A simple linear analysis about an unstable homogeneous equilibrium of the one-dimensional Cahn-Hilliard equation gives heuristic evidence that most solutions that start with initial data near such an equilibrium exhibit a behavior corresponding to spinodal decomposition. In this paper we formulate this conjecture in a mathematically precise way, using geometric and measure-theoretic techniques, and prove its validity. We believe that this is the first rigorous treatment of this phenomenon.