Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions: Nonlinear dynamics (Q1973863)

The authors study the spinodal decomposition for the Cahn-Hilliard equation \[ u_t=-\Delta(\varepsilon^2\Delta u +f(u)), \;\text{ in } \Omega,\qquad {\partial u\over \partial \nu}={\partial\Delta u\over\partial \nu}=0 \;\text{ on } \partial \Omega, \] where \(\Omega\subset \mathbb{R}^n\), \(n\in \{1,2,3\}\), is bounded domain, \(\nu \) is the normal vector field on \(\partial \Omega\), and \(f\) is a cubic like nonlinearity. The equation models the phase separation of a binary alloy and the spinodal decomposition refers to the first stage of this process, in which the mixture of two metallic components becomes inhomogeneous forming a fine-grained structure with a characteristic length scale. The authors offer an interesting mathematical explanation of this phenomenon in terms of solutions of the Cahn-Hilliard equation. They show, roughly speaking, that in any neighborhood of a homogeneous equilibrium, the behavior of almost all solutions is governed by solutions contained in a finite-dimensional attractive invariant manifold, and the spatial structure of the solutions on this manifold is in a good agreement with the structure of the mixture experimentally observed in the spinodal decomposition.