Spectral comparison principles for the Cahn-Hilliard and phase-field equations, and time scales for coarsening

Abstract

We initiate a study of the instability properties of steady solutions of the Cahn-Hilliard (CH) equation and the phase field (PF) system in terms of the coarseness of those solutions. Other spectral results are obtained as well. Steady state solutions for the CH equation with no flux or periodic boundary conditions are also stationary states for the bistable reaction-diffusion equation and the PF system (with constant temperature). Thus, it is meaningful to linearize each of these equations about a common stationary solution and discuss the stability of this state in each dynamical process. Converting the linearized operators to self-adjoint form and expressing eigenvalues in terms of modified Rayleigh quotients enables us to relate the spectra of the three operators. In particular we relate the indices of instability with respect to the three processes. We also discuss the change in the spectrum of the linearized CH operator in one dimension when we change the stationary solution at which it is linearized. We conjecture that the eigenvalues increase with the number of oscillations of the steady state for a given mass, and prove that this is so in certain cases. This lends credence to the idea that motion of these dynamical systems near coarse solutions takes place on longer time scales.